Optimal. Leaf size=54 \[ -\frac {c^3}{2 a^3 x^2}+\frac {5 c^3}{a^2 x}+\frac {11 c^3 \log (x)}{a}-\frac {16 c^3 \log (a x+1)}{a}+c^3 x \]
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Rubi [A] time = 0.14, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 88} \[ -\frac {c^3}{2 a^3 x^2}+\frac {5 c^3}{a^2 x}+\frac {11 c^3 \log (x)}{a}-\frac {16 c^3 \log (a x+1)}{a}+c^3 x \]
Antiderivative was successfully verified.
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Rule 88
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx\\ &=\frac {c^3 \int \frac {e^{-2 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=\frac {c^3 \int \frac {(1-a x)^4}{x^3 (1+a x)} \, dx}{a^3}\\ &=\frac {c^3 \int \left (a^3+\frac {1}{x^3}-\frac {5 a}{x^2}+\frac {11 a^2}{x}-\frac {16 a^3}{1+a x}\right ) \, dx}{a^3}\\ &=-\frac {c^3}{2 a^3 x^2}+\frac {5 c^3}{a^2 x}+c^3 x+\frac {11 c^3 \log (x)}{a}-\frac {16 c^3 \log (1+a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 56, normalized size = 1.04 \[ -\frac {c^3}{2 a^3 x^2}+\frac {5 c^3}{a^2 x}+\frac {11 c^3 \log (a x)}{a}-\frac {16 c^3 \log (a x+1)}{a}+c^3 x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 62, normalized size = 1.15 \[ \frac {2 \, a^{3} c^{3} x^{3} - 32 \, a^{2} c^{3} x^{2} \log \left (a x + 1\right ) + 22 \, a^{2} c^{3} x^{2} \log \relax (x) + 10 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 53, normalized size = 0.98 \[ c^{3} x - \frac {16 \, c^{3} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac {11 \, c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac {10 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 53, normalized size = 0.98 \[ -\frac {c^{3}}{2 x^{2} a^{3}}+\frac {5 c^{3}}{a^{2} x}+c^{3} x +\frac {11 c^{3} \ln \relax (x )}{a}-\frac {16 c^{3} \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 51, normalized size = 0.94 \[ c^{3} x - \frac {16 \, c^{3} \log \left (a x + 1\right )}{a} + \frac {11 \, c^{3} \log \relax (x)}{a} + \frac {10 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 51, normalized size = 0.94 \[ c^3\,x-\frac {\frac {c^3}{2}-5\,a\,c^3\,x}{a^3\,x^2}+\frac {11\,c^3\,\ln \relax (x)}{a}-\frac {16\,c^3\,\ln \left (a\,x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 42, normalized size = 0.78 \[ c^{3} x + \frac {c^{3} \left (11 \log {\relax (x )} - 16 \log {\left (x + \frac {1}{a} \right )}\right )}{a} + \frac {10 a c^{3} x - c^{3}}{2 a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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